A Comprehensive Review on the Generalized Sylvester Equation AX-YB=C

1. Introduction

This review commences with the famous Sylvester equation

$$AX-XB=C$$

with unknown X, named after the British mathematician James Joseph Sylvester (1814–1897) [251]. For a detailed discussion of this equation, refer to the review article [14] by Rajendra Bhatia, a Fellow of the Indian National Science Academy. The core of our review revolves around the generalized Sylvester equation (abbreviated as GSE):

$$AX-YB=C$$

with unknown X and Y.

In 1952, Roth established the necessary and sufficient conditions for the solvability of GSE over a field by means of block matrix equivalence. Since then, a large number of papers on GSE have been published in rapid succession. The solvability conditions and explicit expressions of the general solution for GSE have been investigated from diverse perspectives: linear transformations, generalized inverses, matrix decompositions, real (complex) representations, determinant representations, and semi-tensor products, etc. Extensive studies have been conducted on various constrained solutions (e.g., ★-congruent, symmetric, self-adjoint, positive (semi)definite, per(skew)symmetric, bi(skew)symmetric, Re-(non)negative definite, Re-(non)positive definite, $\eta$-Hermitian, $\eta$-skew-Hermitian, $\varphi$-Hermitian, and equality-constrained solutions) of GSE, as well as its various best approximate solutions when the solvability conditions are not satisfied. Furthermore, the generalizations of GSE in different algebraic structures (e.g., unit regular rings, principal ideal domains, division rings, module-finite rings, commutative rings, Artinian rings, noncommutative rings, dual numbers, dual quaternions), operator equations, tensor equations, polynomial matrix equations, Sylvester-polynomial-conjugate matrix equations, and formal aspects have also shown remarkable vitality. Iterative algorithms for solving various solutions to GSE and its extended forms have also attracted considerable attention, with continuous updates and optimizations. Finally, the relevant results of GSE have demonstrated extraordinary significance in both theoretical applications (e.g., solvability of matrix equations, dual number matrix factorizations, and microlocal triangularization of pseudo-differential systems) and practical applications (e.g., hand-eye calibration problems, and encryption and decryption of color images).

Qing-Wen Wang, the first author of this paper, has been engaged in researching and teaching on the theory of matrix equations since the early 1990s, and has published over 100 related articles. Numerous scholars both domestically and internationally have suggested that we systematically present the theory of solving linear matrix equations. To date, our team has published three review articles, focusing respectively on solving the equations $AXB=C$[284], $AX=C$ and $XB=D$[278], as well as $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$ [279].

As mentioned earlier, in the development history of research on GSE, its conclusions are interrelated, mutually inspiring, mutually promotive, and progressively advanced, forming a complex, intertwined, and extensive network. By reviewing numerous literature, analyzing and comparing, summarizing, questioning, and prospecting, we strive to identify the commonalities, main threads, and approaches in these studies, aiming to provide insights and inspiration for subsequent research on GSE. Thus, this work is of significant value. In this paper, we intend to focus on GSE as the core theme, and  unfold its solution methods, solution categories, generalizations, iterative algorithms, as well as theoretical and practical applications in a step-by-step manner, so as to provide a comprehensive overview.

The remainder of this paper consists of 8 sections. Section 2 presents some necessary notations and notes. In Section 3, we introduce Roth’s work on GSE published in [228], which serves as the starting point for this paper. In Section 4, we summarize seven methods for studying GSE, demonstrating diverse perspectives and research schemes. Various solutions to GSE are discussed in detail in Section 5. We devote Section 6 to introducing the generalizations of GSE in certain algebraic aspects. In Section 7, we enumerate several classic iterative algorithms for solving various solutions to GSE and its generalized forms. The theoretical and practical applications of GSE are presented in Section 8. Conclusions are stated in Section 9.

2. Preliminary

This section recalls some notations and definitions used throughout the paper. Additionally, other necessary terminology for subsequent (sub)sections will be introduced within each. A few symbols, due to their bulk, will be referenced via specific indices in original sources, rather than being reproduced. This not only does not hinder readers’ understanding but also makes the paper more concise and readable.

Let $\mathbb{F}$ be a ring; let $\mathbb{F}\left[\lambda \right]$ be the polynomial ring over $\mathbb{F}$ with the variable $\lambda$; let ${\mathbb{F}}^{m\times n}$ be the set of all $m\times n$ matrices over $\mathbb{F}$; let ${\mathbb{F}}^{m\times n}\left[\lambda \right]$ be the set of all $m\times n$ polynomial matrices over $\mathbb{F}$ with the variable $\lambda$. Specially, ${\mathbb{F}}^m={\mathbb{F}}^{m\times 1}$. Let $degf\left(\lambda \right)$ be the degree of $f\left(\lambda \right)\in \mathbb{F}\left[\lambda \right]$; let $\mathrm{rank}\left(A\right(\lambda \left)\right)$ be the rank of $A\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$; let $A^T$ be the transpose of $A\in {\mathbb{F}}^{m\times n}$. The symbol $det\left(A\right(\lambda \left)\right)$ denotes the determinant of $A\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ for a field $\mathbb{F}$. The component-wise representation of $A\in {\mathbb{F}}^{m\times n}$ can be denoted in the following three forms:

$$A=\left[a_{ij}\right]=\left[a_{i,j}\right]={\left[a_{i,j}\right]}_{m\times n}\in {\mathbb{F}}^{m\times n}.$$

The Kronecker product of $A=\left[a_{ij}\right]\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{s\times t}$ is defined by

$$A\otimes B=\left[\begin{array}{cccc}{a}_{11}B& a_{12}B& \dots & a_{1n}B\\ a_{21}B& a_{22}B& \dots & a_{2n}B\\ ⋮& ⋮& ⋮& ⋮\\ a_{m1}B& a_{m2}B& \dots & a_{mn}B\end{array}\right]\in {\mathbb{F}}^{ms\times nt}.$$

The matrices $A\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{m\times n}$ are equivalent if there exist two nonsingular matrices $P\in {\mathbb{F}}^{m\times m}$ and $Q\in {\mathbb{F}}^{n\times n}$ such that $A=PBQ$. Moreover, $A\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$ are equivalent if there exist two invertible polynomial matrices $P\left(\lambda \right)\in {\mathbb{F}}^{m\times m}\left[\lambda \right]$ and $Q\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ such that $A\left(\lambda \right)=P\left(\lambda \right)B\left(\lambda \right)Q\left(\lambda \right)$.

For a subspace $\mathcal{T}\subseteq {\mathbb{F}}^n$, let $P_{\mathcal{T}}$ be the orthogonal projector onto $\mathcal{T}$; and let ${\mathcal{T}}^{\perp }$ and $dim\left(\mathcal{T}\right)$ be the orthogonal complement and dimension of $\mathcal{T}$, respectively. In addition, denote $A\mathcal{T}=\{At\mid t\in \mathcal{T}\}$ for $A\in {\mathbb{F}}^{m\times n}$. For two vector spaces V and W over a field F, let $\tau$ be a linear transformation from V to W; then $\mathrm{Ker}\left(\tau \right)$ and $\mathrm{Im}\left(\tau \right)$ represent the kernel space and the image of $\tau$, respectively.

Let ${\mathbb{Z}}^{+}$, $\mathbb{N}$, $\mathbb{R}$, and $\mathbb{C}$ be the sets of all positive integers, natural numbers, real numbers, and  complex numbers, respectively. The symbol $\mathrm{sign}\left(a\right)$ stands for the sign of $a\in \mathbb{R}$, and ∅ represents the empty set. Let the set of all (real) quaternions be

$$\mathbb{H}=\{q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}\mid {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,\mathbf{i}\mathbf{j}\mathbf{k}=-1,q_1,q_2,q_3,q_4\in \mathbb{R}\},$$

which is a four-dimensional noncommutative division algebra over $\mathbb{R}$ [227]. Let the conjugate of a quaternion $q=q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}\in \mathbb{H}$ be

$$\overline{q}=q_1-q_2\mathbf{i}-q_3\mathbf{j}-q_4\mathbf{k}.$$

Let $A=\left[a_{ij}\right]\in {\mathbb{H}}^{m\times n}$. The conjugate of A is $\overline{A}=\left[{\overline{a}}_{ij}\right]\in {\mathbb{H}}^{m\times n}$, and  the conjugate transpose of A is $A^{*}={\overline{A}}^T$. Let

$$A^{\eta }=-\eta A\eta \mathit{and}{A^{\eta }}^{*}=-\eta A^{*}\eta ,$$

where $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$. If $A={A^{\eta }}^{*}$ ($A=-{A^{\eta }}^{*}$) for $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$, then A is called $\eta$-Hermitian ($\eta$-anti-Hermitian) [127]. An $\eta$-anti-Hermitian matrix is also called an $\eta$-skew-Hermitian matrix. Moreover, ${A^{\mathbf{i}}}^{*}=A^{*}$ and ${A^{\mathbf{j}}}^{*}={A^{\mathbf{k}}}^{*}=A^T$ for $A\in {\mathbb{C}}^{m\times n}$.

Denote

$$P_A=A^{†}A,Q_A=A{A}^{†},L_A=I-A^{†}A,R_A=I-A{A}^{†},$$

where $A^{†}$ is the Moore-Penrose inverse [215] of a matrix A. The symbols $A^{-1}$, $\mathrm{rank}\left(A\right)$, $\mathcal{R}\left(A\right)$, and $\mathcal{N}\left(A\right)$ denote the inverse, rank, range, and null space of a matrix A, respectively. In addition, $\overline{A}$, $A^{*}$, and ${\parallel A\parallel }_F$ are the conjugate, conjugate transpose, and Frobenius norm of $A\in {\mathbb{C}}^{m\times n}$, respectively. Let I and 0 be the identity matrix and the null matrix, respectively, with  appropriate orders. Specifically, $I_n$ denotes the $n\times n$ identity matrix. For a symmetric matrix $A\in {\mathbb{R}}^{m\times m}$, $A>0$ and $A\ge 0$ denote the positive definite matrix and the positive semidefinite matrix, respectively. In addition, denote $A\ge B$ if $A-B\ge 0$ for matrices A and B. The matrix

$$A=[A_1,A_2,\dots ,A_n]=\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\end{array}\right]$$

denotes a new matrix formed by arranging matrix blocks $A_1,A_2,\dots ,A_n$ (with the same number of rows) column-wise. The symbol $\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_r)$ denotes a (block) diagonal matrix with diagonal entries ${\sigma }_1,\dots ,{\sigma }_r$.

For a ring $\mathbb{F}$ and $a\in \mathbb{F}$, a is regular (or inner-invertible) if there exists $a^{-}\in \mathbb{F}$ such that $a{a}^{-}a=a$; a is $(1,2)$-invertible (or reflexive) if there exists $a^{(1,2)}\in \mathbb{F}$ such that $a{a}^{(1,2)}a=a$ and $a^{(1,2)}a{a}^{(1,2)}=a^{(1,2)}$. The characteristic of a field $\mathbb{F}$ is denoted as $\mathrm{char}\left(\mathbb{F}\right)$.

An equation or a system of equations is said to be solvable (consistent) if it has at least one solution. The symbols “⇒" and “⇔" denote “imply" and “if and only if", respectively.

At the end of this section, we present three specific notes regarding this review.

(1)

Though some research does not focus directly on GSE itself, its traces are easily detectable. Thus, we regard such content as an integral part of GSE-related research.

(2)

We selected core literature relevant to this review. Results closely related to the theme are rigorously presented as theorems, while less relevant conclusions are briefly summarized narratively. Furthermore, the proofs of these theorems are omitted here.

(3)

The remarks in this paper include comments and suggestions on relevant results, encompassing both previous researchers’ views and our reflections, questions, and prospects.

3. Roth’s Equivalence Theorem

First, we specifically present the paper’s core equation:

$$AX-YB=C,$$

(1)

where $A=\left[a_{i,j}\right]\in {\mathbb{F}}^{m\times r}$, $B=\left[b_{i,j}\right]\in {\mathbb{F}}^{s\times n}$, and $C=\left[c_{i,j}\right]\in {\mathbb{F}}^{m\times n}$ are given over a ring $\mathbb{F}$.

Let $\mathbb{F}$ be a field. In 1952, Roth [228] first studied the necessary and sufficient conditions for the solvability to the polynomial matrix form of Eq. (1), i.e.,

$$A\left(\lambda \right)X\left(\lambda \right)-Y\left(\lambda \right)B\left(\lambda \right)=C\left(\lambda \right),$$

(2)

where $A\left(\lambda \right),B\left(\lambda \right),C\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$, by using the normal form of polynomial matrices.

Theorem 1. 

[228] Eq. (2) has a solution pair $X\left(\lambda \right),Y\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ if and only if the polynomial matrices

$$\left[\begin{array}{cc}A\left(\lambda \right)& C\left(\lambda \right)\\ 0& B\left(\lambda \right)\end{array}\right]\mathit{and}\left[\begin{array}{cc}A\left(\lambda \right)& 0\\ 0& B\left(\lambda \right)\end{array}\right]$$

are equivalent.

Roth [228], Theorem 1 has stated that Theorem 1 still valid for rectangular polynomial matrices of appropriate orders. Thus, the following theorem can be derived immediately.

Theorem 2. 

[228], Theorem 1 Let $A\in {\mathbb{F}}^{m\times r}$, $B\in {\mathbb{F}}^{s\times n}$, and $C\in {\mathbb{F}}^{m\times n}$. Then,

$$\mathrm{Eq}.\left(1\right)\mathit{has}a\mathit{solution}\mathit{pair}X\in {\mathbb{F}}^{r\times n}\mathit{and}Y\in {\mathbb{F}}^{m\times s}$$

(3)

if and only if

$$\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]\mathit{and}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\mathit{are}\mathit{equivalent}.$$

(4)

We call this theorem Roth’s equivalence theorem (abbreviated as RET).

Remark 1. 

It is easy to find that the study of Eq. (1) is equivalent to that of the equation

$$AX+YB=C$$

(5)

with given $A=\left[a_{i,j}\right]\in {\mathbb{F}}^{m\times r}$, $B=\left[b_{i,j}\right]\in {\mathbb{F}}^{s\times n}$, and $C=\left[c_{i,j}\right]\in {\mathbb{F}}^{m\times n}$, by regarding $-B$ in Eq. (1) as B in Eq. (5). Thus, in this paper, we collectively refer to Eqs. (1) and (5) as GSE.

4. Different Methods on GSE

Roth [228] considered Eq. (1) based on the canonical forms of the polynomial matrices. Subsequently, many scholars have provided different proofs of RET from various perspectives. These proofs demonstrate the effectiveness and uniqueness of different mathematical methods in considering Eq. (1). In addition, these methods have also had a profound impact on the subsequent study of different Sylvester-type equations. This is precisely one of the enchantment of mathematics: reaching the same endpoint through different paths, or even completely unrelated ones. Furthermore, mathematicians ceaselessly pursue groundbreaking innovations and perpetually strive to discover the most elegant path.

Remark 2. 

It is notable that it is easy to see that (3) implies (4) by

$$\left[\begin{array}{cc}I& -Y\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]=\left[\begin{array}{cc}A& AX-YB\\ 0& B\end{array}\right].$$

Therefore, it is only need to consider the sufficiency of RET, i.e.,  (4) ⇒ (3).

4.1. Method by Linear Transformations and Subspace Dimensions

Let $\mathbb{F}$ be a field. Flanders and Wimmer [80] proved RET for $m=r$ and $s=n$ by means of linear transformations and subspace dimension arguments. This method is more fundamental and elementary than Roth’s method.

Proof of RET. [80]

Step 1: 

Define ${\psi }_i:M_{r+s,2(r+s)}\to M_{r+s}$ by

$$\begin{array}{c}\hfill {\psi }_0(U,W)=\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]U-W\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\mathit{and}{\psi }_1(U,W)=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]U-W\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right].\end{array}$$

Then, the condition (4) yields

$$dim\left(\mathrm{Ker}\left({\psi }_0\right)\right)=dim\left(\mathrm{Ker}\left({\psi }_1\right)\right).$$

Step 2: 

Let

$$U=\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{21}& U_{22}\end{array}\right]\mathit{and}W=\left[\begin{array}{cc}{W}_{11}& W_{12}\\ W_{21}& W_{22}\end{array}\right].$$

Then,

$$\begin{array}{cc}\hfill & \mathrm{Ker}\left({\psi }_0\right)=\left\{(U,W)\left|\begin{array}{cc}A{U}_{11}=W_{11}A,\hfill & A{U}_{12}=W_{12}B,\hfill \\ B{U}_{21}=W_{21}A,\hfill & B{U}_{22}=W_{22}B,\hfill \end{array}\right.\right\},\hfill \\ \hfill & \mathrm{Ker}\left({\psi }_1\right)=\left\{(U,W)\left|\begin{array}{cc}A{U}_{11}+C{U}_{21}=W_{11}A,& A{U}_{12}+C{U}_{22}=W_{12}B,\\ B{U}_{21}=W_{21}A,& B{U}_{22}=W_{22}B,\end{array}\right.\right\}.\hfill \end{array}$$

Let

$$Z=\left\{\left[\begin{array}{cccc}{U}_{21}& U_{22}& W_{21}& W_{22}\end{array}\right]\mid B{U}_{21}=W_{21}A,B{U}_{22}=W_{22}B\right\}.$$

For $i=0,1$, define

$${\nu }_i:\mathrm{Ker}\left({\psi }_i\right)\to Z,{\nu }_i(U,W)=\left[\begin{array}{cccc}{U}_{21}& U_{22}& W_{21}& W_{22}\end{array}\right].$$

Then $\mathrm{Im}\left({\nu }_1\right)\subseteq \mathrm{Im}\left({\nu }_0\right)=Z\mathit{and}\mathrm{Ker}\left({\nu }_1\right)=\mathrm{Ker}\left({\nu }_0\right).$ So, $\mathrm{Im}\left({\nu }_1\right)=\mathrm{Im}\left({\nu }_0\right)$.

Step 3: 

Since $(U,W)\in Ker\left({\psi }_0\right)$ with $U_{22}=-I$, there also exists such in $\mathrm{Ker}\left({\psi }_1\right)$. Therefore, $A{U}_{12}-W_{12}B=C$, i.e.,  (3) holds. □

Remark 3. 

(1)

In [80], Flanders and Wimmer mentioned that by making small modifications to the above proof, one can similarly obtain the proof of RET under the condition of rectangular matrices A, B, and C.

(2)

In terms of linear transformations and subspace dimensions, Dmytryshyn et al. discussed two highly complex systems of equations (see [59], Theorem 1.1 and [58], Theorem 1).

4.2. Method by Generalized Inverses

Let $\mathbb{F}$ be a field. In 1955, Penrose [215] concisely defined the Moore-Penrose inverse of any rectangular matrix through four matrix equations.

Definition 1. 

[9,215] Let $A\in {\mathbb{F}}^{m\times n}$. If there exists $X\in {\mathbb{F}}^{n\times m}$ such that

$$\begin{array}{c}\hfill \left(1\right)AXA=A,\left(2\right)XAX=X,\left(3\right){\left(AX\right)}^{*}=AX,\left(4\right){\left(XA\right)}^{*}=XA,\end{array}$$

then X is called the Moore-Penrose (in short, MP) inverse of A and is denoted by $A^{†}$. Especially, if  $X\in {\mathbb{F}}^{n\times m}$ satisfies the above Eq. $\left(1\right)$, then X is called an inner inverse of A and is denoted by $A^{-}$. Clearly, $A^{†}$ is a special inner inverse of A.

Since then, the theory of generalized inverses has flourished (see [9,40,223,265]), and it has been widely used to study the solvability conditions of linear equations and represent the explicit expressions of their general solutions when solvable. Penrose’s study on the matrix equation $AXB=C$ [215] is the most renowned.

In 1979, Baksalary and Kala [6] naturally utilized inner inverses to establish solvability conditions and representations of the general solution for Eq. (1).

Theorem 3. 

[6] Eq. (1) has a solution pair $X\in {\mathbb{F}}^{r\times n}$ and $Y\in {\mathbb{F}}^{m\times s}$ if and only if

$$\left(I-A{A}^{-}\right)C\left(I-B^{-}B\right)=0,$$

(6)

in which case,

$$\begin{array}{cc}\hfill X& =A^{-}C+A^{-}ZB+\left(I-A^{-}A\right)W,\hfill \\ \hfill Y& =-\left(I-A{A}^{-}\right)C{B}^{-}+Z-\left(I-A{A}^{-}\right)ZB{B}^{-},\hfill \end{array}$$

where W and Z are arbitrary matrices with appropriate orders.

Remark 4. 

In views of RET and Theorem 3, Baksalary and Kala [6], Remark 2 noted that

$$(6)\iff (4)\iff \mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right).$$

They also pointed out that this result can be directly derived by using [197], Fomula (8.7) in a particular case, i.e.,

$$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right)+\mathrm{rank}\left((I-A{A}^{-})C(I-B^{-}B)\right).$$

Moreover, they said that proof using generalized inverses is simpler than Roth’s [228] and Flanders et al.’s [80]. In fact, this is indeed the case when considering the simplicity of the proof and its length.

Remark 5. 

Under the hypotheses of Theorem 3, it is easy to obtain that

$$\begin{array}{cc}\hfill \left(6\right)& \iff \mathcal{R}\left(C\left(I-B^{-}B\right)\right)\subseteq \mathcal{N}\left(\left(I-A{A}^{-}\right)\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill \iff C\mathcal{N}\left(B\right)\subseteq \mathcal{R}\left(A\right),\end{array}$$

(7)

which is also proved by Woude in [260], Lemma 3.2 according to an elementary method. In addition, Woude applied this result to a control problem that occurs in almost non-stationary stochastic processes via measurement feedback (see [260], Theorems 3.3 and 4.1).

Remark 6. 

It is easy to show that the solvability of Eq. (1) is equivalent to the existence of X and Y such that

$$\left[\begin{array}{cc}I& -Y\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right].$$

(8)

In terms of a geometrical method, Olshevsky gave a cyclic argument over $\mathbb{C}$ as follows:

$$\left(3\right)\Rightarrow \left(8\right)\Rightarrow \left(4\right)\Rightarrow \left(7\right)\Rightarrow \left(3\right),$$

(see [209], Proof of Theorem 1.2).

Let $\mathbb{F}=\mathbb{C}$. Meyer [201] revealed an interesting result, that is, the solvability of Eq. (1) is equivalent to the existence of the upper block inner inverse of a certain block matrix.

Theorem 4. 

[201], Theorems 1 and 2 The block triangular complex matrix

$$T=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]$$

has an upper block triangular inner inverse if and only if

$$\mathrm{rank}\left(T\right)=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right),$$

in which case,

$$T^{-}=\left[\begin{array}{cc}{A}^{-}& -A^{-}C{B}^{-}\\ 0& B^{-}\end{array}\right],$$

is an inner inverse of T for any $A^{-}$ and $B^{-}$.

4.3. Method by Singular Value Decompositions

Let $\mathbb{F}=\mathbb{R}$. The singular value decomposition (abbreviated as SVD) [89], as an important tool for the study of matrix theory, also plays a significant role in the research on solutions of matrix equations. Chu [34] utilized the SVD to study the solvability conditions and the representation of the general solution of Eq. (5).

In fact, let the SVDs of A and B given in Eq. (5) be

$$A=U_A{D}_A{V}_A^T\mathit{and}B=U_B{D}_B{V}_B^T,$$

(9)

where $U_A$, $U_B$, $V_A$, and  $V_B$ are orthogonal matrices with appropriate orders,

$$D_A=\left[\begin{array}{cc}{\Sigma }_A& 0\\ 0& 0\end{array}\right],D_B=\left[\begin{array}{cc}{\Sigma }_B& 0\\ 0& 0\end{array}\right],$$

${\Sigma }_A=\mathrm{diag}({\alpha }_1,..,{\alpha }_k)$ with ${\alpha }_i>0$ ($1\le i\le k$), and ${\Sigma }_B=\mathrm{diag}({\delta }_1,..,{\delta }_l)$ with ${\delta }_j>0$ ($1\le i\le l$). Then,

$$\left(5\right)\iff U_A{D}_A{V}_A^{T}X+Y{U}_B{D}_B{V}_B^T=C\iff D_A\tilde{X}+\tilde{Y}{D}_B=\tilde{C},$$

where $\tilde{X}=V_A^{T}X{V}_B$, $\tilde{Y}=U_A^{T}Y{U}_B$, and  $\tilde{C}=U_A^{T}C{V}_B$. Partitioning the above equation analogously to the partitioning of $D_A$ and $D_B$, we obtain

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}{\Sigma }_A& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}{\tilde{X}}_{11}& {\tilde{X}}_{12}\\ {\tilde{X}}_{21}& {\tilde{X}}_{22}\end{array}\right]+\left[\begin{array}{cc}{\tilde{Y}}_{11}& {\tilde{Y}}_{12}\\ {\tilde{Y}}_{21}& {\tilde{Y}}_{22}\end{array}\right]\left[\begin{array}{cc}{\Sigma }_B& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}{\tilde{C}}_{11}& {\tilde{C}}_{12}\\ {\tilde{C}}_{21}& {\tilde{C}}_{22}\end{array}\right]\hfill \\ \hfill \iff & \left[\begin{array}{cc}{\Sigma }_A{\tilde{X}}_{11}+{\tilde{Y}}_{11}{\Sigma }_B& {\Sigma }_A{\tilde{X}}_{12}\\ {\tilde{Y}}_{21}{\Sigma }_B& 0\end{array}\right]=\left[\begin{array}{cc}{\tilde{C}}_{11}& {\tilde{C}}_{12}\\ {\tilde{C}}_{21}& {\tilde{C}}_{22}\end{array}\right].\hfill \end{array}$$

(10)

Based on the above discussion, the following theorem can be obtained.

Theorem 5. 

[34], Theorem 2 Under the notations in (9) and (10), let

$$U_A=\left[\begin{array}{cc}{U}_{A_1}& U_{A_2}\end{array}\right]\mathit{and}{V}_B=\left[\begin{array}{cc}{V}_{B_1}& V_{B_2}\end{array}\right].$$

(1)

Then Eq. (5) is solvable if and only if

$${\tilde{C}}_{22}=0,\mathrm{i}.\mathrm{e}.,U_{A_2}^{T}C{V}_{B_2}=0.$$

(11)

(2)

Suppose that (11) holds. Denote

$$\tilde{C}=\left({\tilde{c}}_{ij}\right)\mathit{and}{M}_{ij}=\left[\begin{array}{cc}{\alpha }_i& {\delta }_j\end{array}\right].$$

Then ${\tilde{X}}_{21}$, ${\tilde{X}}_{22}$, ${\tilde{Y}}_{12}$, and ${\tilde{Y}}_{22}$ are arbitrary,

$${\tilde{X}}_{12}={\Sigma }_A^{-1}{\tilde{C}}_{12},{\tilde{Y}}_{21}={\tilde{C}}_{21}{\Sigma }_B^{-1},{\tilde{X}}_{11}=\left({\tilde{x}}_{ij}\right),\mathit{and}{\tilde{Y}}_{11}=\left({\tilde{y}}_{ij}\right),$$

where

$$\left[\begin{array}{c}{\tilde{x}}_{ij}\\ {\tilde{y}}_{ij}\end{array}\right]=M_{ij}^{†}{\tilde{c}}_{ij}+(I-M_{ij}^{†}{M}_{ij})Z_{ij}$$

for arbitrary $Z_{ij}$. Moreover, if  ${\tilde{Y}}_{11}$ is arbitrary, then

$${\tilde{X}}_{11}={\Sigma }_A^{-1}({\tilde{C}}_{11}-{\tilde{Y}}_{11}{\Sigma }_B).$$

(3)

If (11) holds, then

$$\begin{array}{cc}\hfill X& =V_A\left[\begin{array}{cc}{\Sigma }_A^{-1}(U_{A_1}^{T}C{V}_{B_1}-Z_1{\Sigma }_B)& {\Sigma }_A^{-1}{U}_{A_1}^{T}C{V}_{B_1}\\ Z_2& Z_3\end{array}\right]V_B^T,\hfill \\ \hfill Y& =U_A\left[\begin{array}{cc}{Z}_1& Z_4\\ U_{A_2}^{T}C{V}_{B_1}{\Sigma }_B^{-1}& Z_5\end{array}\right]U_B^T,\hfill \end{array}$$

where $Z_1,...,Z_5$ are arbitrary matrices with appropriate orders.

Remark 7. 

Interestingly, Chu in [34], Theorem 1 utilized the generalized singular value decomposition (abbreviated as GSVD) [212] to study the extended form of Eq. (5) over $\mathbb{R}$, i.e.,

$$AXE+FYB=C,$$

(12)

where $A,E,F,B$, and C are given real matrices with appropriate orders. Eleven years later, Xu et al. [323] once again discussed the solvability conditions, the general solution, and least-squares solutions of Eq. (12) over $\mathbb{C}$ by using the canonical correlation decomposition (abbreviated as CCD) introduced in [90].

Remark 8. 

Inspired by solving Eq. (5) via SVD, one can utilize equivalent normal forms of A and B to study Eq. (5) over a field, along with an approach similar to Theorem 5. In fact, we only need to regard orthogonal matrices $U_A$, $U_B$, $V_A$, and  $V_B$ in (9) as invertible matrices, the transpose $V_A^T$ and $V_B^T$ as the inverse $V_A^{-1}$ and $V_B^{-1}$, and ${\Sigma }_A$ and ${\Sigma }_B$ as identity matrices of appropriate orders.

It can be found that using the equivalent normal form of matrices to solve matrix equations is also a very convenient method. This viewpoint has been confirmed multiple times in the subsequent research. Wang et al. used the elementary row and column transformations of matrices to give the equivalent normal form of a matrix triplet

$$\left[\begin{array}{ccc}A& B& C\end{array}\right]$$

with the same row number over an arbitrary division ring $\mathbb{F}$ (see [273], Theorem 2.1). Similarly, the equivalent normal form of another matrix triplet

$${\left[\begin{array}{ccc}{D}^T& E^T& F^T\end{array}\right]}^T$$

with the same column number can also be obtained (see [273], Theorem 2.2). The two equivalent normal forms are applied to solve the matrix equation

$$AXD+BYE+CZF=G,$$

(13)

where X, Y, and Z are unknown (see [273], Theorem 3.2).

Interestingly, He et al. utilized [273], Theorem 2.1 to propose a simultaneous decomposition of seven matrices over $\mathbb{H}$ (see [125], Theorem 2.3), i.e.,

$$\left[\begin{array}{cccc}G& A& B& C\\ D& & & \\ E& & & \\ F& & \end{array}\right]$$

and discussed Eq. (13) once again by this decomposition (see [125], Theorem 3.1). Compared with the method in [273], which directly applies the equivalent normal forms of two matrix triplets to matrix G, the  simultaneous decomposition is more concise. In addition, Ref. [114] once again brilliantly demonstrates the important role of simultaneous decomposition in solving matrix equations.

4.4. Method by Simultaneous Decompositions

Let $\mathbb{F}$ be a field. Remark 8 notes using equivalent canonical forms of the matrices A and B to solve Eq. (1). Can we find an equivalent canonical form that is simultaneously related to the matrix triplet

$$\left[\begin{array}{cc}C& A\\ B\end{array}\right]$$

(14)

so as to discuss Eq. (1) more simply? Gustafson [97] gave a positive answer to this problem.

Theorem 6. 

[97] Let A, B, and C be given in Eq. (1). Then there exist invertible matrices T, U, V, and  W of appropriate orders such that

$$\left[\begin{array}{cc}TCW& TAU\\ VBW\end{array}\right]=\left[\begin{array}{cc}{C}^{\prime }& A^{\prime }\\ B^{\prime }\end{array}\right],$$

(15)

where

$$A^{\prime }=\left[\begin{array}{cccc}{I}_z& 0& 0& 0\\ 0& I_{t_3}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& I_{r_2}\\ 0& 0& 0& 0\end{array}\right],B^{\prime }=\left[\begin{array}{cccccc}{I}_z& 0& 0& 0& 0& 0\\ 0& I_{t_1}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0& 0\end{array}\right],C^{\prime }=\left[\begin{array}{cccccc}{I}_z& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0& 0\\ 0& 0& 0& 0& I_{r_2}& 0\\ 0& 0& 0& 0& 0& I_{t_2}\end{array}\right],$$

$z+t_3+r_2=\mathrm{rank}\left(A\right)$, $z+t_1+r_1=\mathrm{rank}\left(B\right)$, and  $z+t_2+r_1+r_2=\mathrm{rank}\left(C\right)$. We call (15) the simultaneous decomposition of (14).

Remark 9. 

By elementary matrix row and column transformations, Gustafson designed the 12-step algorithm to obtain the simultaneous decomposition of (14) (see [97], Section 3).

Applying the simultaneous decomposition of (14) to Eq. (1) yields

$$\begin{array}{cc}\hfill \left(1\right)& \iff T^{-1}{A}^{\prime }{U}^{-1}X-Y{V}^{-1}{B}^{\prime }{W}^{-1}=T^{-1}{C}^{\prime }{W}^{-1}\hfill \\ \hfill & \iff A^{\prime }{X}^{\prime }-Y^{\prime }{B}^{\prime }=C^{\prime },\hfill \end{array}$$

(16)

where $X^{\prime }=U^{-1}XW$ and $Y^{\prime }=TY{V}^{-1}$. Thus, $X=U{X}^{\prime }{W}^{-1}$ and $Y=T^{-1}{Y}^{\prime }V$.

Theorem 7. 

[97] Eq. (16) is solvable if and only if $t_2=0$, in which case,

$$A^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& I_{t_3}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_2}& 0\\ 0& 0& 0& 0& 0\end{array}\right],B^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& I_{t_1}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0\end{array}\right],C^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0\\ 0& 0& 0& 0& I_{r_2}\end{array}\right],$$

$$X^{\prime }=\left[\begin{array}{ccccc}{X}_{11}& X_{12}& 0& X_{14}& 0\\ X_{21}& X_{22}& 0& X_{24}& 0\\ X_{31}& X_{32}& X_{33}& X_{35}& X_{35}\\ X_{41}& X_{42}& 0& X_{44}& I_{r_2}\end{array}\right],Y^{\prime }=\left[\begin{array}{cccc}{X}_{11}-I_z& X_{12}& Y_{13}& X_{14}\\ X_{21}& X_{22}-I_{t_1}& Y_{23}& X_{24}\\ 0& 0& Y_{33}& 0\\ 0& 0& Y_{43}& -I_{r_1}\\ X_{41}& X_{42}& Y_{53}& X_{44}\end{array}\right],$$

where the $X_{ij}$ and $Y_{ij}$ are arbitrary matrices with appropriate orders.

Remark 10. 

The quiver theory introduced by Gabriel [84] is an important tool in the representation theory of algebras. Gustafson [97] gave a novel interpretation of the existence of the simultaneous decomposition of (14) from the perspective of the quiver theory. Finally, he gave a necessary and sufficient condition for Eq. (1) to have a solution by using the corresponding representation of arrows (see [97], Section 10). Noted that Refs. [59,115] also make use of the graph theory to discuss linear matrix equations.

Remark 11. 

Wang  et al. in [276], Theorem 2.1 continued with the idea of simultaneous decomposition and decomposed the following matrices over a division ring $\mathbb{F}$:

$$\left[\begin{array}{ccc}A& B& C\\ D& \end{array}\right],$$

(17)

where A, B, and C are of the same row number, and A and D are of the same column number. So, Theorem 6 is a corollary of [276], Theorem 2.1 (see [276], Corollary 2.2). Also, it should be noted that [273], Theorem 2.1 is indeed a special case of [276], Theorem 2.1 (see [276], Corollary 2.3). In addition, Wang et al. applied the simultaneous decomposition of (17) to solving two types of systems of matrix equations. Interestingly, He et al. [116] further refined the simultaneous decomposition presented by [276].

He et al. [126] further considered the simultaneous decompositions of two more general forms over $\mathbb{H}$:

$$\left[\begin{array}{c}A\\ B& C& D\\ & & E\end{array}\right]\mathit{and}\left[\begin{array}{c}A\\ B& C\\ & D\\ & E\end{array}\right],$$

where matrices in the same row (or column) have the same number of rows (or columns), and  applied them to solving systems of quaternion matrix equations. Recently, Huo  et al. [135] generalized the simultaneous decomposition of (17) to quaternion tensors under the Einstein product.

4.5. Method by Real (Complex) Representations

It is well known that a associative algebra $\mathcal{A}$ with finite dimensions over a field $\mathbb{F}$ is isomorphic to a subalgebra of the algebra ${\mathbb{F}}^{n\times n}$, where n is the dimension of $\mathcal{A}$ over $\mathbb{F}$ (see [81]). We now consider the case of complex numbers, i.e., $\mathbb{F}=\mathbb{C}$.

Let $A=A_0+A_1\mathbf{i}\in {\mathbb{C}}^{m\times r}$, where $A_0,A_1\in {\mathbb{R}}^{m\times r}$, and  $\mathbf{i}$ is the imaginary unit such that ${\mathbf{i}}^2=-1$. Define a map

$$\varphi :{\mathbb{C}}^{m\times r}\to {\mathbb{R}}^{2m\times 2r}\mathrm{with}\varphi \left(A\right)=\varphi (A_0+A_1\mathbf{i})=\left[\begin{array}{cc}{A}_0& -A_1\\ A_1& A_0\end{array}\right].$$

We call $\varphi \left(A\right)$ a real representation of the complex matrix A[191]. Then, one can check that $\varphi (·)$ is an isomorphism of the real algebra ${\mathbb{C}}^{m\times n}$ onto the real subalgebra

$$\left\{\left[\begin{array}{cc}{S}_0& -S_1\\ S_1& S_0\end{array}\right]\mid S_0,S_1\in {\mathbb{R}}^{m\times r}\right\}.$$

Then,

$$\left(5\right)\iff \varphi \left(A\right)\varphi \left(X\right)+\varphi \left(Y\right)\varphi \left(B\right)=\varphi \left(C\right).$$

On the other hand, suppose that a real matrix pair

$$\widehat{X}=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ X_{21}& X_{22}\end{array}\right]\mathit{and}\widehat{Y}=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]$$

is a solution pair of the following equation

$$\varphi \left(A\right)\widehat{X}+\widehat{Y}\varphi \left(B\right)=\varphi \left(C\right).$$

(18)

Then, $K_{2r}^{-1}\widehat{X}{K}_{2n}$ and $K_{2m}^{-1}\widehat{Y}{K}_{2s}$ are also a solution pair of Eq. (18), where

$$K_{2t}=\left[\begin{array}{cc}0& I_t\\ -I_t& 0\end{array}\right]\mathrm{for}t=m,n,r,s.$$

Let $\overline{X}={\displaystyle \frac{1}{2}}(X_{11}+X_{22})+{\displaystyle \frac{1}{2}}(X_{21}-X_{12})\mathbf{i}$ and $\overline{Y}={\displaystyle \frac{1}{2}}(Y_{11}+Y_{22})+{\displaystyle \frac{1}{2}}(Y_{21}-Y_{12})\mathbf{i}$. Then,

$$\begin{array}{cc}\hfill \varphi \left(\overline{X}\right)& ={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{X}_{11}+X_{22}& X_{12}-X_{21}\\ X_{21}-X_{12}& X_{11}+X_{22}\end{array}\right]={\displaystyle \frac{1}{2}}\left(\widehat{X}+K_{2r}^{-1}\widehat{X}{K}_{2n}\right),\hfill \\ \hfill \mathit{and}\varphi \left(\overline{Y}\right)& ={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{Y}_{11}+Y_{22}& Y_{12}-Y_{21}\\ Y_{21}-Y_{12}& Y_{11}+Y_{22}\end{array}\right]={\displaystyle \frac{1}{2}}\left(\widehat{Y}+K_{2m}^{-1}\widehat{Y}{K}_{2s}\right)\hfill \end{array}$$

satisfy (18), so, $\overline{X}$ and $\overline{Y}$ satisfy Eq. (5).

Based on the above analysis, the problem of a complex matrix equation is transformed into a problem of a real matrix equation by using the real representation of complex matrices. Based on this consideration, Liu [191] proposed the following theorem for solving Eq. (5) over $\mathbb{C}$.

Theorem 8. 

[191], Lemma 1.3 Let

$$A=A_0+A_1\mathbf{i},B=B_0+B_1\mathbf{i},\mathit{and}C=C_0+C_1\mathbf{i}$$

be given in Eq. (5). Then, Eq. (5) is consistent over $\mathbb{C}$ if and only if

$$\left[\begin{array}{cc}{A}_0& -A_1\\ A_1& A_0\end{array}\right]\left[\begin{array}{cc}{X}_{11}& X_{12}\\ X_{21}& X_{22}\end{array}\right]+\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]\left[\begin{array}{cc}{B}_0& -B_1\\ B_1& B_0\end{array}\right]=\left[\begin{array}{cc}{C}_0& -C_1\\ C_1& C_0\end{array}\right]$$

(19)

is consistent over $\mathbb{R}$, in which case,

$$\left\{\begin{array}{c}X=X_0+X_1\mathbf{i}={\displaystyle \frac{1}{2}}(X_{11}+X_{22})+{\displaystyle \frac{1}{2}}(X_{21}-X_{12})\mathbf{i},\hfill \\ Y=Y_0+Y_1\mathbf{i}={\displaystyle \frac{1}{2}}(Y_{11}+Y_{22})+{\displaystyle \frac{1}{2}}(Y_{21}-Y_{12})\mathbf{i},\hfill \end{array}\right.$$

(20)

where $X_{11},X_{12},X_{21},X_{22},Y_{11},Y_{12},Y_{21}$, and $Y_{22}$ are the general solution of Eq. (19). Furthermore, the explicit forms of X and Y given in (20) are

$$\begin{array}{cc}\hfill X_0=& {\displaystyle \frac{1}{2}}{P}_1\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_1+{\displaystyle \frac{1}{2}}{P}_2\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_2+[U_1,U_2]\left[\begin{array}{c}\varphi \left(B\right)Q_1\\ \varphi \left(B\right)Q_2\end{array}\right]+[P_1{F}_{\varphi \left(A\right)},P_2{F}_{\varphi \left(A\right)}]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right],\hfill \\ \hfill X_1=& {\displaystyle \frac{1}{2}}{P}_2\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_1-{\displaystyle \frac{1}{2}}{P}_1\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_2+[U_1,U_2]\left[\begin{array}{c}-\varphi \left(B\right)Q_2\\ \varphi \left(B\right)Q_1\end{array}\right]+[P_2{F}_{\varphi \left(A\right)},-P_1{F}_{\varphi \left(A\right)}]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right],\hfill \\ \hfill Y_0=& {\displaystyle \frac{1}{2}}{S}_1{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_1+{\displaystyle \frac{1}{2}}{S}_2{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_2-[S_1\varphi \left(A\right),S_2\varphi \left(A\right)]\left[\begin{array}{c}{\widehat{U}}_1\\ {\widehat{U}}_2\end{array}\right]+[W_1,W_2]\left[\begin{array}{c}{E}_{\varphi \left(B\right)}{T}_1\\ E_{\varphi \left(B\right)}{T}_2\end{array}\right],\hfill \\ \hfill Y_1=& {\displaystyle \frac{1}{2}}{S}_2{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_1-{\displaystyle \frac{1}{2}}{S}_1{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_2+[-S_2\varphi \left(A\right),S_1\varphi \left(A\right)]\left[\begin{array}{c}{\widehat{U}}_1\\ {\widehat{U}}_2\end{array}\right]+[W_1,W_2]\left[\begin{array}{c}{E}_{\varphi \left(B\right)}{T}_1\\ -E_{\varphi \left(B\right)}{T}_2\end{array}\right],\hfill \end{array}$$

where $P_1=[I_r,0],P_2=[0,I_r],S_1=[I_m,0],S_2=[0,I_m]$,

$$\begin{array}{cc}\hfill Q_1& =\left[\begin{array}{c}{I}_n\\ 0\end{array}\right],Q_2=\left[\begin{array}{c}0\\ I_n\end{array}\right],T_1=\left[\begin{array}{c}{I}_s\\ 0\end{array}\right],T_2=\left[\begin{array}{c}0\\ I_s\end{array}\right],\hfill \end{array}$$

and $U_i,V_i,{\widehat{U}}_i,W_i(i=1,2)$ are arbitrary matrices over $\mathbb{R}$ with appropriate orders.

Remark 12. 

Liu [191] further used Theorem 8 to discuss the maximal and minimal ranks of Eq. (5)’s solutions, as seen in SubSection 5.6 of this paper.

Remark 13. 

We know that the quaternion algebra (or the quaternion division ring) cannot be a field because it does not satisfy the commutative law. However, when studying matrix equations over $\mathbb{H}$, it is widely applied that the method of converting the quaternion matrix equations to the real (or complex) matrix equations by using the real (or complex) representation of quaternions [227]. For example, [337] and [333] studied the special least-squares solutions of a class of quaternion matrix equations by using the real representation and the complex representation of quaternions, respectively. Moreover, the real (or complex) representation of some other quaternions mentioned in Remark 43 have been explored in [190,226,264,332].

On the other hand, because real (or complex) representations have the drawback of a significant increase in computational load, Wei et al. [292] introduced real structure-preserving methods over $\mathbb{H}$ for LU decomposition, QR decomposition, and SVD, thereby solving quaternion linear systems.

4.6. Method by Determinantal Representations

As is well known, Cramer’s rule for the linear equation $Ax=b$ for unknown vector x is an effective means of expressing its unique solution. In addition, it can be found in SubSection 4.2 that the theory of generalized inverses is closely related to the study of Eq. (1). Naturally, we can consider whether Cramer’s rule for the solutions of Eq. (1) can be obtained through determinant representations of generalized inverses.

Generalized inverses (especially the MP inverse) over fields (particularly the complex field) have been thoroughly discussed and applied to characterize various solutions of matrix equations (see [9,249,265]). Notably, Kyrchei [169] presented the determinantal representations of the MP inverse and the Drazin inverse [62] from the new perspective of limit representations of generalized inverses in 2008.

When we consider the determinantal representations of generalized inverses in the quaternion algebra, it relies on the theory of determinants of quaternion matrices. However, due to the noncommutativity of quaternions, the determinants of quaternion matrices become much more complicated (see [4,37]). It was not until several decades later, after Kyrchei [168,171] introduced the theory of column-row determinants over $\mathbb{H}$, that this problem was effectively solved.

Definition 2 

(Column-row determinants over $\mathbb{H}$). [168], Definitions 2.4 and 2.5 Let $A=\left[a_{ij}\right]\in {\mathbb{H}}^{n\times n}$, and let $S_n$ be the symmetric group on $I_n=\{1,2,\dots ,n\}$.

(1)

For $i=1,2,\dots ,n$, the i-th row determinant of A is defined by

$$\begin{array}{cc}\hfill {\mathrm{rdet}}_{i}A& =\sum _{\sigma \in S_n}{(-1)}^{n-r}(a_{i{i}_{k_1}}{a}_{i_{k_1}{i}_{k_1+1}}\dots a_{i_{k_1+l_1}i})\dots (a_{i_{k_r}{i}_{k_r+1}}\dots a_{i_{k_r+l_r}{i}_{k_r}}),\hfill \\ \hfill \sigma & =(i{i}_{k_1}{i}_{k_1+1}\dots i_{k_1+l_1})(i_{k_2}{i}_{k_2+1}\dots i_{k_2+l_2})\dots (i_{k_r}{i}_{k_r+1}\dots i_{k_r+l_r}),\hfill \end{array}$$

where $i_{k_2}<i_{k_3}<\dots <i_{k_r}$, and $i_{k_t}<i_{k_t+s}$ for $t=2,\dots ,r$ and $s=1,\dots ,l_t$.

(2)

For $j=1,2,\dots ,n$, the j-th column determinant of A is defined by

$$\begin{array}{cc}\hfill {\mathrm{cdet}}_{j}A& =\sum _{\tau \in S_n}{(-1)}^{n-r}(a_{j_{k_r}{j}_{k_r+l_r}}\dots a_{j_{k_r+1}{j}_{k_r}})\dots (a_{j{j}_{k_1+l_1}}\dots a_{j_{k_1+1}{j}_{k_1}}{a}_{j_{k_1}j}),\hfill \\ \hfill \tau & =(j_{k_r+l_r}\dots j_{k_r+1}{j}_{k_r})\dots (j_{k_2+l_2}\dots j_{k_2+1}{j}_{k_2})(j_{k_1+l_1}\dots j_{k_1+1}{j}_{k_1}j),\hfill \end{array}$$

where $j_{k_2}<j_{k_3}<\dots <j_{k_r}$ and $j_{k_t}<j_{k_t+s}$ for $t=2,\dots ,r$ and $s=1,\dots ,l_t$.

Remark 14. 

Kyrchei in [168], Theorem 3.1 showed that if $A\in {\mathbb{H}}^{n\times n}$ is Hermitian, i.e., $A=A^{*}$, then

$${\mathrm{rdet}}_{1}A=\dots ={\mathrm{rdet}}_{n}A={\mathrm{cdet}}_{1}A=\dots ={\mathrm{cdet}}_{n}A\in \mathbb{R}.$$

Thus, [168], Remark 3.1 defines the determinant of a Hermitian matrix A by

$$detA={\mathrm{rdet}}_{i}A={\mathrm{cdet}}_{i}A,(i=1,2,\dots ,n).$$

We now introduce some necessary notations. Let $A\in {\mathbb{H}}^{m\times n}$. Then, ${\mathcal{R}}_l\left(A\right)$, ${\mathcal{R}}_r\left(A\right)$, ${\mathcal{N}}_l\left(A\right)$, and  ${\mathcal{N}}_r\left(A\right)$ denote the left row space, the right column space, the left null space, and  the right null space of A, respectively. Let

$$\alpha =\{{\alpha }_1,\dots ,{\alpha }_k\}\subseteq \{1,\dots ,m\}\mathit{and}\beta =\{{\beta }_1,\dots ,{\beta }_k\}\subseteq \{1,\dots ,n\},$$

where $1\le k\le min\{m,n\}$. By $A_{\beta }^{\alpha }$, we denote the submatrix of A whose rows are indexed by $\alpha$ and whose columns are indexed by $\beta$. If A is Hermitian, then $|A_{\alpha }^{\alpha }|$ is the corresponding principal minor of A. For $1\le k\le n$, let

$$L_{k,n}=\{\alpha \mid \alpha =({\alpha }_1,\dots ,{\alpha }_k),1\le {\alpha }_1<\dots <{\alpha }_k\le n\}.$$

And, for  $i\in \alpha$ and $j\in \beta$, let

$$I_{r,m}\left\{i\right\}=\{\alpha \mid \alpha \in L_{r,m},i\in \alpha \}\mathit{and}{J}_{r,n}\left\{j\right\}=\{\beta \mid \beta \in L_{r,n},j\in \beta \}.$$

We denote the j-th column of A by ${\mathbf{a}}_{.j}$ and its i-th row by ${\mathbf{a}}_{i.}$. And, $A_{.j}\left(\mathbf{b}\right)$ is the matrix that is obtained from A by replacing its j-th column by the column vector $\mathbf{b}\in {\mathbb{H}}^{m\times 1}$, and $A_{i.}\left(\mathbf{b}\right)$ is the matrix obtained from A by replacing its i-th row by the row vector $\mathbf{b}\in {\mathbb{H}}^{1\times n}$. Symbols ${\mathbf{a}}_{·j}^{*}$ and ${\mathbf{a}}_{i·}^{*}$ denoted the j-th column and the i-th row of $A^{*}$, respectively.

Kyrchei [173] obtained the Cramer’s rules for some left, right and two-sided quaternion matrix equations by the theory of the column-row determinants over $\mathbb{H}$. Song et al. [246] then utilized results in [173] to further study the Cramer’s rule for the quaternion matrix equation:

$$AXE+FYB=C,$$

(21)

where $A\in {\mathbb{H}}^{m\times n}$, $E\in {\mathbb{H}}^{s\times q}$, $F\in {\mathbb{H}}^{m\times q}$, $B\in {\mathbb{H}}^{t\times q}$, and $C\in {\mathbb{H}}^{m\times q}$ are given.

Theorem 9. 

[246], Theorem 3.1 Suppose that Eq. (21) is consistent. Let

$$\begin{array}{cc}\hfill & T=A^{*}(I+R_F)A,S=E(I+L_B)E^{*},A_{11}=A^{*}{R}_{F}A{T}^{†},A_{22}=A^{*}A{T}^{†},\hfill \\ \hfill & E_{11}=S^{†}E{E}^{*},E_{22}=S^{†}E{L}_B{E}^{*},Y_{10}=A_{22}^{†}{A}_{11}{A}^{*}C{L}_B{E}^{*}+L_{A_{22}}{A}^{*}{R}_{F}C{E}^{*}{E}_{22}{E}_{11}^{†},\hfill \\ \hfill & Y_{20}=A_{11}^{†}{A}_{22}{A}^{*}{R}_{F}C{E}^{*}+L_{A_{11}}{A}^{*}C{L}_B{E}^{*}{E}_{11}{E}_{22}^{†}.\hfill \end{array}$$

Let $K^{*}$, L, $M^{*}$, and N be of full column rank matrices over $\mathbb{H}$ such that

$${\mathcal{N}}_r\left(T\right)={\mathcal{R}}_r\left(K^{*}\right),{\mathcal{N}}_r\left(S\right)={\mathcal{R}}_r\left(L\right),{\mathcal{N}}_r\left(F\right)={\mathcal{R}}_r\left(M^{*}\right),{\mathcal{N}}_r\left(B\right)={\mathcal{R}}_r\left(N\right).$$

Then, the general solution of Eq. (21) is

$$\begin{array}{cc}\hfill x_{ij}& ={\displaystyle \frac{{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(c_{i.}^A\right)}{\det (T+K^{*}K)\det (S+L{L}^{*})}}={\displaystyle \frac{{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(c_{.j}^E\right)}{\det (T+K^{*}K)\det (S+L{L}^{*})}},\hfill \\ \hfill y_{hl}& ={\displaystyle \frac{{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(g_{h.}^A\right)}{\det (F^{*}F+M^{*}M)\det (B{B}^{*}+N{N}^{*})}}={\displaystyle \frac{{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(g_{.l}^E\right)}{\det (F^{*}F+M^{*}M)\det (B{B}^{*}+N{N}^{*})}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill c_{i.}^A& =\left[{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(d_{.1}\right),\dots ,{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(d_{.s}\right)\right],\hfill \\ \hfill g_{h.}^A& =\left[{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(k_{.1}\right),\dots ,{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(k_{.t}\right)\right],\hfill \\ \hfill c_{.j}^E& ={\left[{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(d_{1.}\right),\dots ,{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(d_{n.}\right)\right]}^T,\hfill \\ \hfill g_{.l}^E& ={\left[{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(k_{1.}\right),\dots ,{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(k_{q.}\right)\right]}^T\hfill \end{array}$$

with $d_{i.}$ and $d_{.j}$ are the i-th row and j-th column of

$$(T+K^{*}K)T^{†}(A^{*}{R}_{F}C{E}^{*}+A^{*}C{L}_B{E}^{*}+Y_{10}+Y_{20})S^{†}(S+L{L}^{*})+M+W,$$

respectively, and  $k_{.i}$ and $k_{j.}$ are the i-th column and j-th row of

$$(F^{*}F+M^{*}M)\left(F^{†}(C-AXE)B^{†}\right)(B{B}^{*}+N{N}^{*})+Q,$$

respectively, for $i=1,\dots ,m$, $j=1,\dots ,s$, $h=1,\dots ,q$, and $l=1,\dots ,t$, where

$$\begin{array}{cc}\hfill M& =(T+K^{*}K)T^{†}(L_{A_{22}}{V}_1{R}_{E_{11}}+L_{A_{11}}{V}_2{R}_{E_{22}})S^{†}(S+L{L}^{*}),\hfill \\ \hfill W& =TZL{L}^{*}+K^{*}KZS+K^{*}KZL{L}^{*},Q=F^{*}FHN{N}^{*}+M^{*}MH(B{B}^{*}+N{N}^{*})\hfill \end{array}$$

for arbitrary $V_1$, $V_2$, Z, and H with appropriate orders.

Remark 15. 

When E and F in Eq. (21) are identity matrices, by Theorem 9 we immediately derived the Cramer’s rule to the matrix equation over $\mathbb{H}$:

$$AX+YB=C.$$

(22)

Note that Theorem 9 uses the auxiliary matrices, i.e., K, L, M, and N, to derive the determinant representation of the general solution of Eq. (22). However, these auxiliary matrices are not always easy to obtain in practical applications. Then, can we obtain the Cramer’s rule for Eq. (22) only through its given coefficient matrices?

In order to answer the above problem, let’s first take a look at another work of Kyrchei [170]. Based on the column-row determinant theory and the limit representation of the MP inverse, he gave the determinant representation of the MP inverse over $\mathbb{H}$ in [170]. This research has greatly promoted the study of the Cramer’s rules to the various solutions of matrix equations over $\mathbb{H}$ (see [164,165,166,172,243,244,245]). Denote

$${\mathbb{H}}_r^{m\times n}=\{A\in {\mathbb{H}}^{m\times n}|\mathrm{rank}\left(A\right)=r\}.$$

Theorem 10. 

[170], Theorem 5 Let $A\in {\mathbb{H}}_r^{m\times n}$ and $A^{†}=\left[a_{i,j}^{+}\right]\in {\mathbb{H}}^{n\times m}$. Then,

$$a_{i,j}^{+}={\displaystyle \frac{{\sum }_{\beta \in J_{r,n}\left\{i\right\}}{\mathrm{cdet}}_i{\left({\left(A^{*}A\right)}_{·i}\left({\mathbf{a}}_{·j}^{*}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r,n}}\left|{\left(A^{*}A\right)}_{\beta }^{\beta }\right|}}={\displaystyle \frac{{\sum }_{\alpha \in I_{r,m}\left\{j\right\}}{\mathrm{rdet}}_j{\left({\left(A{A}^{*}\right)}_{j·}\left({\mathbf{a}}_{i·}^{*}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r,m}}\left|{\left(A{A}^{*}\right)}_{\alpha }^{\alpha }\right|}},$$

where $i=1,2.\dots ,n$ and $j=1,2,\dots ,m$.

A result similar to Theorem 3 was given by Kyrchei in [165].

Theorem 11. 

[165], Lemma 5.1 The following are equivalent:

(1)

Eq. (22) is solvable;

(2)

$R_{A}C{L}_B=0$;

(3)

$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]$;

in which case,

$$\begin{array}{c}\hfill X=A^{†}C-A^{†}VB+L_{A}U\mathit{and}Y={\mathrm{R}}_{A}C{B}^{†}+A{A}^{†}V+W{\mathrm{R}}_B,\end{array}$$

(23)

where U, V, and W are arbitrary matrices of appropriate orders over $\mathbb{H}$.

It is easy to see that

$$\begin{array}{c}\hfill X=A^{†}C\mathit{and}Y=C{B}^{†}-A{A}^{†}C{B}^{†}\end{array}$$

(24)

are the partial solution pair of Eq. (22) by taking U, V, and W as zero matrices in (23). Then, by applying Theorem 10 to (24), Kyrchei [165] presented a new Cramer’s rule for the partial solution of Eq. (22), which only makes use of the coefficient matrices, i.e.,  A, B, and C.

Theorem 12. 

[165], Theorem 5.2 Let $A\in {\mathbb{H}}_{r_1}^{m\times n}$ and $B\in {\mathbb{H}}_{r_2}^{t\times q}$. Then, the  partial solution pair $X=\left[x_{ij}\right]\in {\mathbb{H}}^{n\times q}$ and $Y=\left[y_{gf}\right]\in {\mathbb{H}}^{m\times t}$ in (24) can be expressed as

$$x_{ij}={\displaystyle \frac{{\sum }_{\beta \in J_{r_1,n}\left\{i\right\}}{\mathrm{cdet}}_i{\left({\left(A^{*}A\right)}_{.i}\left({\mathbf{c}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_1,n}}\left|{\left(A^{*}A\right)}_{\beta }^{\beta }\right|}},$$

where ${\mathbf{c}}_{.j}^{\left(1\right)}$ is the j-th columns of $A^{*}C$, and 

$$\begin{array}{cc}\hfill y_{gf}& ={\displaystyle \frac{{\sum }_{\alpha \in I_{r_2,t}\left\{f\right\}}{\mathrm{rdet}}_f{\left({\left(B{B}^{*}\right)}_{.f}\left({\mathbf{c}}_{g.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r_2,t}}\left|{\left(B{B}^{*}\right)}_{\alpha }^{\alpha }\right|}}\hfill \\ \hfill & -{\displaystyle \frac{{\sum }_{l=1}^m{\sum }_{\alpha \in I_{r_1,m}\left\{l\right\}}{\mathrm{rdet}}_l{\left({\left(A{A}^{*}\right)}_{l.}\left({\ddot{\mathbf{a}}}_{g.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha }{\sum }_{\alpha \in I_{r_2,t}\left\{f\right\}}{\mathrm{rdet}}_f{\left({\left(B{B}^{*}\right)}_{.f}\left({\mathbf{c}}_{l.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r_1,m}}\left|{\left(A{A}^{*}\right)}_{\alpha }^{\alpha }\right|{\sum }_{\alpha \in I_{r_2,t}}\left|{\left(B{B}^{*}\right)}_{\alpha }^{\alpha }\right|}},\hfill \end{array}$$

where ${\mathbf{c}}_{g.}^{\left(2\right)}$ and ${\ddot{\mathbf{a}}}_{g.}^{\left(1\right)}$ are the g-th rows of $C{B}^{*}$ and $A{A}^{*}$, respectively.

Remark 16. 

Note that although Kyrchei [165] gave the determinant representation of the particular solution of Eq. (22), the problem proposed in Remark 15 is not solved completely. That is to say, the Cramer’s rule for representing the general solution of Eq. (22) only using the coefficient matrices remains an unsolved problem.

Interestingly, Song [247] considered the determinant representation for the general solution of Eq. (22) with the restricted conditions, i.e.,

$$\mathrm{Eq}.\left(22\right)\mathrm{subject}\mathrm{to}\left\{\begin{array}{c}{\mathcal{R}}_r\left(X\right)\subseteq T_1,{\mathcal{N}}_r\left(X\right)\supseteq S_1,\hfill \\ {\mathcal{R}}_l\left(Y\right)\subseteq T_2,{\mathcal{N}}_l\left(Y\right)\supseteq S_2,\hfill \end{array}\right.$$

(25)

where $T_1\subseteq {\mathbb{H}}^n$, $S_1\subseteq {\mathbb{H}}^p$, $T_2\subseteq {\mathbb{H}}^{1\times t}$, and $S_2\subseteq {\mathbb{H}}^{1\times m}$. Let $P_{\mathcal{T}}\in {\mathbb{H}}^{n\times n}$ ($Q_{\mathcal{T}}\in {\mathbb{H}}^{n\times n}$) denote the right (left) $\mathbb{H}$-orthogonal projector onto a right (left) $\mathbb{H}$-vector subspace $\mathcal{T}\in {\mathbb{H}}^{n\times 1}$ ($\mathcal{T}\in {\mathbb{H}}^{1\times n}$ ) along ${\mathcal{T}}^{\perp }$.

Theorem 13. 

[247], Theorem 3.2 Let A, B, C, $T_1$, $T_2$, $S_1$, and  $S_2$ be given in (25), and let

$$M=R_{A{P}_{T_1}}{Q}_{S_2^{\perp }},N=Q_{T_2}B{P}_{S_1^{\perp }},T=A^{*}{R}_{Q_{S_2^{\perp }}}A+A^{*}A,S=I+L_{Q_{T_2}B},$$

and $Y_1$ be such that

$$A^{*}{R}_{Q_{S_2^{\perp }}}A{T}^{†}{Y}_1=A^{*}A{T}^{†}{A}^{*}{R}_{Q_{S_2^{\perp }}}C\mathit{and}{Y}_1{L}_{Q_{T_2}B}=A^{*}C{L}_{Q_{T_2}B}.$$

(1)

The restricted Eq. (25) is solvable if and only if

$$\begin{array}{c}\hfill R_M{R}_{A{P}_{T_1}}C=0,R_{A{P}_{T_1}}C{L}_{Q_{T_2}B}=0,C{L}_{P_{S_1^{\perp }}}{L}_N=0,R_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}=0,\end{array}$$

in which case,

$$\begin{array}{cc}\hfill X& ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1}{L}_{A{P}_{T_1}}{U}_1{P}_{S_1^{\perp }}+P_{T_1}{V}_1{P}_{S_1^{\perp }}\hfill \\ \hfill & ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1}{Z}_1{P}_{S_1^{\perp }}-P_{T_{11}}{Z}_1{P}_{S_1^{\perp }}\hfill \\ \hfill & ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1\cap {\mathcal{N}}_r\left(A\right)}{U}_2{P}_{S_1^{\perp }}+P_{T_1}{V}_2{P}_{S_1^{\perp }},\hfill \\ \hfill Y& =Q_{S_2^{\perp }}(C-AX){\left(Q_{T_2}B\right)}^{†}+Q_{S_2^{\perp }}{V}_3{R}_{Q_{T_2}B}{Q}_{T_2},\hfill \end{array}$$

where

$$\tilde{C}=\left(A^{*}{R}_{Q_{S_2^{\perp }}}C+A^{*}C{L}_{Q_{T_2}B}+A^{*}{R}_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}+Y_1\right)S^{-1},$$

and $Z_1$, $V_1$, $U_1$, $V_2$, $U_2$, and $V_3$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

(2)

Let $C_1^{*},K_1^{*},C_1$, and $K_2$ be full column rank matrices such that

$$T_1={\mathcal{N}}_r\left(C_1\right),T_1\cap {\mathcal{N}}_r\left(T\right)={\mathcal{R}}_r\left(K_1^{*}\right),T_2={\mathcal{N}}_l\left(C_2\right),T_2\cap {\mathcal{N}}_r\left(B\right)={\mathcal{R}}_l\left(K_2^{*}\right).$$

Denote $X=\left[x_{ij}\right]\in {\mathbb{H}}^{n\times q}$ and $Y=\left[y_{kl}\right]\in {\mathbb{H}}^{m\times t}$. If Eq. (25) is solvable, then

$$\begin{array}{cc}\hfill x_{ij}& ={\displaystyle \frac{{\mathrm{cdet}}_i{(A^{*}A+C_1^{*}{C}_1+K_1^{*}{K}_1)}_{.i}\left(d_{.j}\right)}{det(A^{*}A+C_1^{*}{C}_1+K_1^{*}{K}_1)}},\hfill \\ \hfill y_{kl}& ={\displaystyle \frac{{\mathrm{rdet}}_l{(B{B}^{*}+C_2{C}_2^{*}+K_2{K}_2^{*})}_{l.}\left(d_{k.}\right)}{det(B{B}^{*}+C_2{C}_2^{*}+K_2{K}_2^{*})}}\hfill \end{array}$$

with $d_{.j}$ is the j-th column of

$$T\left(A^{*}{R}_{Q_{S_2^{\perp }}}C+A^{*}C{L}_{Q_{T_2}B}+A^{*}{R}_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}+Y_1\right)+K_1^{*}{K}_1{Z}_1,$$

and $d_{k.}$ is the i-th row of

$$(C-AX)B^{*}+Z_2{K}_2{K}_2^{*},$$

where $i=1,\dots ,n$, $j=1,\dots ,q$, $k=1,\dots ,m$, $l=1,\dots ,t$, and  $Z_1$ and $Z_2$ are arbitrary matrices over $\mathbb{H}$ with appropriate orders.

4.7. Method by Semi-Tensor Prodcuts

To effectively handle multidimensional arrays and nonlinear problems, Chinese mathematician Daizhan Cheng [30] pioneered a new matrix product in 2001: the semi-tensor product (abbreviated as STP). It exactly coincides with traditional matrix products when the two factor matrices meet the dimension requirements. Due to the excellent properties of STP [26], namely:

(1)

It is applied to any two matrices;

(2)

It has certain commutative properties;

(3)

It inherits all properties of the conventional matrix product;

(4)

It enables easy expression of multilinear functions (mappings);

STP has been effectively applied to: Boolean (control) networks, logical dynamic systems, systems biology, graph theory, formation control, finite automata and symbolic dynamics, circuit design and failure detection, coding and cryptography, fuzzy control, engineering, game theory, and so on (see [25,26,27,31,32] and references therein).

Definition 3. 

[30] Let $\mathbb{F}$ be a ring, $A\in {\mathbb{F}}^{m\times n}$, and  $B\in {\mathbb{F}}^{p\times q}$. The left STP of A and B is defined as

$$A⋉B=\left(A\otimes I_{{\displaystyle \frac{t}{n}}}\right)\left(B\otimes I_{{\displaystyle \frac{t}{p}}}\right),$$

where $t=\mathrm{lcm}(n,p)$ is the least common multiple of n and p.

Remark 17. 

Similarly, Cheng [25] defined the right SPT of A and B, i.e.,

$$A⋊B=\left(I_{{\displaystyle \frac{t}{n}}}\otimes A\right)\left(I_{{\displaystyle \frac{t}{p}}}\otimes B\right).$$

However, its mathematical properties are not as good as those of the left STP in certain aspects, accounting for most studies’ focus on the left STP. Therefore, the STP referred to in this paper specifically denotes the left STP.

Remark 18. 

As a generalization of the traditional matrix product, and given its extensive application significance, the study of matrix equations under STP has emerged as an inevitable and pivotal research direction. In 2016, Yao et al. [331] were the first to conduct research on the classical complex matrix equation under STP:

$$A⋉X=B,$$

where X is unknown. Subsequently, building on the research framework of [331], investigations into diverse matrix equations under STP have flourished (see [137,144,180,266,268,271]).

Regrettably, no studies directly addressing Eq. (5) under STP have been published to date. However, when the dimension of Y is constrained to equal that of $X^T$, the exploration of Sylvester-transpose matrix equation [137]

$$A⋉X+X^T⋉B=C$$

with unknown X may provide valuable guidance for this problem.

Notably, in 2019, Cheng and Liu [29], inspired by the research on cross-dimensional linear systems [28], further proposed the second matrix-matrix semi-tensor product (abbreviated as MM-2 STP), denoted by ${\circ }_l$. Subsequently, Wang [267] investigated the complex matrix equation under the MM-2 STP:

$$A{\circ }_{l}X=B,$$

where X is unknown, and A and B are given. So, exploring Eq. (1) under MM-2 STP emerges as a potential research topic. Additionally, Cheng [26] introduced the dimension-free matrix theory, which systematically elucidates the deep mathematical insights underlying STP.

Since 2020, Liaocheng University’s team led by Professors Ying Li and Jianli Zhao has utilized STP to propose several novel matrix representations, including complex, quaternion, and octonion matrices. These representations have also been applied to solving corresponding matrix equations, and have yielded effective numerical experimental results.

Ding et al. [55] first proposed the real vector representation for quaternion matrices, whose properties were characterized by STP, as follows:

Definition 4. 

[55], Definitions 3.1-3.3 Let $x=x_1+x_2\mathbf{i}+x_3\mathbf{j}+x_4\mathbf{k}\in \mathbb{H}$. Denote

$$v^R\left(x\right)={\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\end{array}\right]}^T.$$

Let $x=\left[\begin{array}{ccc}{x}^1& \dots & x^n\end{array}\right]\in {\mathbb{H}}^{1\times n}$ and $y={\left[\begin{array}{ccc}{y}^1& \dots & y^n\end{array}\right]}^T\in {\mathbb{H}}^n$. Denote

$$v^R\left(x\right)=\left[\begin{array}{c}{v}^R\left(x^1\right)\\ ⋮\\ v^R\left(x^n\right)\end{array}\right]\mathit{and}{v}^R\left(y\right)=\left[\begin{array}{c}{v}^R\left(y^1\right)\\ ⋮\\ v^R\left(y^n\right)\end{array}\right].$$

For $A\in {\mathbb{H}}^{m\times n}$, the real column stacking form $v_c^R\left(A\right)$ and the real row stacking form $v_r^R\left(A\right)$ of A are defined as

$$v_c^R\left(A\right)=\left[\begin{array}{c}{v}^R\left({\mathrm{Col}}_1\left(A\right)\right)\\ v^R\left({\mathrm{Col}}_2\left(A\right)\right)\\ ⋮\\ v^R\left({\mathrm{Col}}_n\left(A\right)\right)\end{array}\right]\mathit{and}{v}_r^R\left(A\right)=\left[\begin{array}{c}{v}^R\left({\mathrm{Row}}_1\left(A\right)\right)\\ v^R\left({\mathrm{Row}}_2\left(A\right)\right)\\ ⋮\\ v^R\left({\mathrm{Row}}_m\left(A\right)\right)\end{array}\right],$$

where ${\mathrm{Col}}_j\left(A\right)(j=1,...,n)$ and ${\mathrm{Row}}_i\left(A\right)(j=1,...,m)$ are the j-th column and i-th row of A, respectively.

Remark 19. 

For $A\in {\mathbb{H}}^{m\times n}$ and $B\in {\mathbb{H}}^{n\times p}$,55], Theorem 3.3(3) shows

$$v_r^R\left(AB\right)=G\left(v_r^R\left(A\right)⋉v_c^R\left(B\right)\right),$$

where

$$\begin{array}{cc}\hfill G& =\left[\begin{array}{c}F⋉{\left({\delta }_m^1\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^1\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^1\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^p\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^m\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^1\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^m\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^p\right)}^T\right]\end{array}\right],F=M_Q⋉\left(\sum _{i=1}^n{\left({\delta }_n^i\right)}^T⋉(I_{4n}\otimes {\left({\delta }_n^i\right)}^T)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill M_Q& =\left[\begin{array}{cccccccccccccccc}1& 0& 0& 0& 0& -1& 0& 0& 0& 0& -1& 0& 0& 0& 0& -1\\ 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& -1& 0\\ 0& 0& 1& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0& 1& 0& 0& -1& 0& 0& 1& 0& 0& 0\end{array}\right],\hfill \end{array}$$

and ${\delta }_m^i(i=1,...,m)$ is the i-th column of $I_m$.

Using the real vector representation for quaternion matrices, Ding et al. [55] further discussed the special least-squares solutions of quaternion matrix equation

$$AXB+CYD=E,$$

(26)

where $A\in {\mathbb{H}}^{m\times n}$, $B\in {\mathbb{H}}^{n\times s}$, $C\in {\mathbb{H}}^{m\times k}$, $D\in {\mathbb{H}}^{k\times s}$, and  $E\in {\mathbb{H}}^{m\times s}$. Denote

$${\delta }_k[i_1,\dots ,i_s]=[{\delta }_k^{i_1},\dots ,{\delta }_k^{i_s}],$$

$$W_{[m,n]}={\delta }_{mn}[1,\dots ,(n-1)m+1,\dots ,m,\dots ,nm],$$

and

$$J^{\eta }=\left[\begin{array}{c}{J}_1^{\eta }\\ ⋮\\ J_m^{\eta }\\ ⋮\\ J_n^{\eta }\end{array}\right],J_m^{\eta }=\left[\begin{array}{c}{J}_{1m}^{\eta }\\ ⋮\\ J_{rm}^{\eta }\\ ⋮\\ J_{nm}^{\eta }\end{array}\right],R^{\eta }=\left[\begin{array}{c}{R}_1^{\eta }\\ ⋮\\ R_m^{\eta }\\ ⋮\\ R_n^{\eta }\end{array}\right],R_m^{\eta }=\left[\begin{array}{c}{R}_{1m}^{\eta }\\ ⋮\\ R_{rm}^{\eta }\\ ⋮\\ R_{nm}^{\eta }\end{array}\right],m=1,2,\dots ,n,$$

where

$$\begin{array}{cc}\hfill & \mathrm{for}\eta =\mathbf{i},J_{rm}^{\mathbf{i}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes R_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.R_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{j},J_{rm}^{\mathbf{j}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes L_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.L_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{k},J_{rm}^{\mathbf{k}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes S_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.S_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{for}\eta =\mathbf{i},R_{rm}^{\mathbf{i}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes R_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.R_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{j},R_{rm}^{\mathbf{j}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes L_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.L_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{k},R_{rm}^{\mathbf{k}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes S_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.S_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

Theorem 14. 

[55], Theorem 4.3 and Corollary 4.4 Let A, B, C, D, and E be given in Eq. (26), and let

$$\widehat{M}=\left[\begin{array}{cc}{M}_1& M_2\end{array}\right],$$

where

$$\begin{array}{cc}\hfill M_1& =G_2⋉G_3⋉v_r^R\left(A\right)⋉W_{[4ns,4n^2]}⋉v_c^R\left(B\right)⋉J^{\eta },\hfill \\ \hfill M_2& =G_4⋉G_5⋉v_r^R\left(C\right)⋉W_{[4ks,4k^2]}⋉v_c^R\left(D\right)⋉R^{\eta },\hfill \end{array}$$

and $G_i$ has the same structure as G given in Remark 19 but differs in orders.

(1)

Then, Eq. (26) is consistent if and only if

$$\left(\widehat{M}{\widehat{M}}^{†}-I_{4ms}\right)v_r^R\left(E\right)=0.$$

(2)

Let

$$S_M=\left\{(X,Y)|X={X^{\eta }}^{*},Y=-{Y^{\eta }}^{*},\parallel AXB+CYD-E\parallel =min\right\}.$$

Then,

$$S_M=\left\{(X,Y)|\left[\begin{array}{c}{v}_s^R\left(X\right)\\ v_s^R\left(Y\right)\end{array}\right]={\widehat{M}}^{†}{v}_r^R\left(E\right)+\left(I_{2(n^2+k^2)+2(n+k)}-{\widehat{M}}^{†}\widehat{M}\right)y,y\in {\mathbb{R}}^{2(n^2+k^2)+2(n+k)}\right\}.$$

(3)

If $(\widehat{X},\widehat{Y})\in S_M$ satisfies

$$\parallel \widehat{X}{\parallel }^2+{\parallel \widehat{Y}\parallel }^2=\underset{(X,Y)\in S_M}{min}\left({\parallel X\parallel }^2+{\parallel Y\parallel }^2\right),$$

then

$$\left[\begin{array}{c}{v}_s^R\left(\widehat{X}\right)\\ v_s^R\left(\widehat{Y}\right)\end{array}\right]={\widehat{M}}^{†}{v}_r^R\left(E\right).$$

Remark 20. 

Setting B and C in Eq. (26) as identity matrices, Theorem 14 yields the corresponding results for Eq. (5) over $\mathbb{H}$.

Remark 21. 

In the same period, Ding et al. [57] and Wang et al. [263] also developed the real vector representation of quaternion matrices and applied this method to address problems in quaternion matrix equations. Inspired by this idea, Liu et al. [189] proposed the left and right real element representations of octonion matrices to solve the classical octonion matrix equation

$$AXB=C,$$

where X is unknown. Recently, Chen and Song [23] used the real vector representation of quaternion matrices to study the least-squares lower (upper) triangular Toeplitz solutions and (anti)centrosymmetric solutions of quaternion matrix equations.

Remark 22. 

It is known that real representations of quaternion matrices are not unique. For instance, Liu et al. [188] defined three real representations of quaternion matrices. Interestingly, Fan et al. first [70,71] proposed the $\mathcal{L}$-representation for quaternion matrices by STP to systematically study the real representations of quaternion matrices. Moreover, $\mathcal{L}$-representation serves as an effective tool in solving quaternion matrix equations. Meanwhile, Zhang et al. [339] also defined the $\mathcal{L}$-representation for commutative quaternion matrices and applied it to solving the corresponding matrix equations.

Following this idea, Fan et al. [72] established the $\mathcal{C}$-representation of quaternion matrices, which generalizes the complex representation of quaternion matrices. This new representation is also applied to study η-Hermitian solutions of the quaternion matrix equation

$$AXB=C,$$

with unknown X. Similarly, Xi et al. [316] defined the ${\mathcal{L}}_C$-representation of reduced biquaternion matrices to investigate mixed solutions of the reduced biquaternion matrix equation

$$\sum _{i=1}^n{A}_i{X}_i{B}_i=E,$$

(27)

where $X_i(i=1,...,n)$ is unknown. Evidently, Eq. (5) is the special case of Eq. (27) over reduced biquaternions.

Contemporaneously with [70,71], Fan et al. [73] also derived the minimal norm least-squares (anti)-Hermitian solution of the quaternion matrix equation directly by the vectorization properties of STP (i.e., [73], Theorems 2.9 and 2.10) and the complex representation of quaternion matrix. Recently, Liu et al. [192] directly use the vectorization properties of STP to investigate (skew) bisymmetric solutions of the generalized Lyapunov quaternion matrix equation.

We contend that the four aforementioned STP-based methods—specifically, $\mathcal{L}$-representation, $\mathcal{C}$-representation, ${\mathcal{L}}_{\mathcal{C}}$-representation, and vectorization properties of STP—provide distinct perspectives for the investigation of Eq. (1).